Electron correlations in charge coupled vertically stacked quantum rings

Transcription

1 PHYSICAL REVIEW B 75, Electron correlations in charge coupled vertically stacked quantum rings B. Szafran, S. Bednarek, and M. Dudziak Faculty of Physics and Applied Computer Science, AGH University of Science and Technology, Aleja Mickiewicza 3, 3-59 Kraków, Poland Received 3 February 7; published June 7 We study spatial correlations for two electrostatically coupled electrons in a pair of vertically stacked quantum rings with configuration-interaction approach. We demonstrate that for distant quantum rings, the correlations undergo abrupt oscillations with the external magnetic field, which are strong for odd multiples of the flux quantum for which the electrons distinctly avoid being localized one above the other and are negligible for other values of the flux. The oscillations of the correlation strength vanish in the single-ring limit for which the electrons form a Wigner molecule. The wave function of the Hartree approximation is frustrated between reproducing the exact angular momentum eigenstate or the correlation between the electrons. We show that for distant rings the mean-field solution breaks the rotational symmetry of the system in a reentrant manner near odd multiples of half of the flux quantum to account for the correlation appearing in the exact solution. DOI:.3/PhysRevB PACS number s : 73..La I. INTRODUCTION Electrons confined in quantum dots exhibit strongly correlated properties when the interaction energy is large compared to the single-electron quantization which occurs in large structures or at high magnetic fields. The correlations are also significant in quantum rings, 3 where an approximate degeneracy of the ground state with respect to the angular momentum is accompanied by the particularly strong character of a nearly one-dimensional Coulomb repulsion, leading to the appearance of the fractional Aharonov-Bohm AB period. Recent fabrication of double concentric quantum rings 5 as well as vertically stacked rings motivated a number of theoretical studies 7 on the properties of carriers confined in these complexes. In this work, we investigate the correlation between the electrons localized in a pair of vertically stacked rings that arise from the mutual Coulomb coupling. A strong correlation is likely to appear due to the reentrant degeneracy of the single-electron ground state in the AB angular momentum oscillations. 3 We study the correlations using the configuration-interaction method, and the exact solution is confronted with the results of the Hartree approximation. Mean-field approaches to the strongly correlated problems are useful in revealing the essential features of the correlation. For circular quantum dots in the fractional quantum Hall regime, the current spin density-functional theory 5 and Hartree-Fock approaches yield broken-symmetry solutions with laboratory-frame Wigner crystallization accounting for the particularly strong tendency of the electrons to avoid one another. Another broken-symmetry density-functional solution 7 for high magnetic field maps the nucleation of vortices bound to electrons as formation of holes in the charge-density droplet. In large one-dimensional dots, the Wigner crystallization is accompanied by antiferromagnetic spin ordering in the internal degrees of freedom which, in the spin local-density method, is mapped as the symmetrybreaking spin-density wave 9 in the laboratory frame of reference. Below, we demonstrate that the exact approach predicts an appearance of the two-electron ground states, in which the interring correlations acquire an infinite range, i.e., persist for arbitrarily large interring distances. For these states, the mean-field method results in the symmetry breaking that is reentrant with the magnetic flux. A critical interring distance exists below which the mean-field solution breaks the symmetry for an arbitrary flux. The insight in the correlation aspects that can be described by the Hartree method is also useful for the density-functional approaches applied recently for the quantum ring complexes,3 since it forms the technical basis on which the local-density methods are constructed. In quantum rings the energy spacings between the angular momentum eigenstates is much smaller than between the radial excitations, and we are mainly interested in the correlation that arise via the electrostatic coupling between the rings for large interring distance. In that case, the electron correlation is bound to have an angular character. Therefore, for clarity of the picture, we choose to work in the strictly onedimensional model of the ring. The model neglects the tunnel coupling between the rings and implies that the electrons are perfectly separated. The electron separation is a reasonable assumption for structures in which the tunnel coupling is not strong, and is exactly realized for large interring barrier for which the main message of the paper is obtained. On the other hand, for strong interring tunnel coupling, the electrons will react on the perpendicular magnetic field like an electron couple in a single ring. The model of stacked onedimensional rings reproduces the single-ring limit for small ring separation in spite of the neglected tunneling see Ref. and the results below. The paper is organized as follows: Section II describes the theory applied for the exact solution of the problem, the results are given in Sec. III exact solution in Sec. III A, mean field in Sec. III B. Summary and conclusions are given in Sec. IV. II. THEORY Hamiltonian of a single electron inside a strictly onedimensional quantum ring of radius R pierced by the magnetic flux takes the form 9-/7/75 3 / The American Physical Society

2 SZAFRAN, BEDNAREK, AND DUDZIAK h = i / e / /mr, where m is the effective electron band mass we adopt GaAs value m=.7m and is the angular position of the electron. The eigenfunctions of Eq. are the angular momentum eigenstates f l =exp il /, with the energy eigenvalues E l = l / /mr, where = /e is the magnetic flux quantum. Hamiltonian for the electron couple is H = h + h + e / r, with standing for the dielectric constant =.9 is used for calculations. Hamiltonian is diagonalized using the configuration-interaction approach with the basis assuming that the electrons occupy different rings r,r = a l l u l,z, d l,z, l,l + b l l d l,z, u l,z,, 3 l,l where a l,l and b l,l are the linear variational parameters; u l and d l are single-electron eigenfunctions of angular momentum l that are localized in the upper and lower rings, respectively, u l = R z Z/ exp il / and dl = R z+z/ exp il /, where stands for a normalized function that is strongly localized at the origin. Note that u l r = l d l r. For the two electrons residing in rings spaced vertically by Z contained within ±Z/ planes, the interelectron distance is given by r =R Z /R + cos, where and stand for the angular coordinates of the two electrons. The system discussed is characterized by two meaningful parameters: the ratio of the distance between the rings and their radius, which is responsible for the impact of the relative angle between the electrons on the Coulomb potential, and the radius R which determines the ratio of the single-electron quantization and the interaction energy. We will consider two different values of R: 3. nm of the order corresponding to the self-assembled quantum rings, which represent the strong confinement case with dominating single-electron quantization and 3 nm the weak confinement case with dominating electron-electron interaction, corresponding to quantum rings produced by the surface oxidation technique. Hamiltonian commutes with the total angular momentum and the parity r r operators. Moreover, the spatial wave functions of the spin eigenstates can be symmetric spin singlets or antisymmetric spin triplets with respect to the electron interchange. Due to the perfect electron separation, degeneracies with respect to the parity and the spin appear and the only relevant quantum number is the total angular momentum. The degeneracy with respect to the parity results from the assumed absence of the interring tunnel coupling and the spin degeneracy from no overlap between the wave functions of the two electrons. We will explain more specifically the degeneracies for the case of the weak interaction limit large Z when the dominant configuration can easily be indicated by the lowest kinetic energy. Namely, PHYSICAL REVIEW B 75, for weak interaction, the wave functions of lowest-energy even angular momentum L eigenstates are constructed of single-electron wave functions carrying angular momentum L ± L = u L r d L r ± d L r u L r /, where L + L corresponds to the spin singlet triplet and is an even- odd- parity eigenfunction. Energy levels for both wave functions L ± will be degenerate since the interaction matrix elements vanish d rd * L r u L r u * L r d L r /r =, 5 the integrand being zero everywhere. The states of odd angular momentum L + for weak interaction will be constructed from L and L + angular momentum eigenfunctions. Using g =u L+ r d L r, g =u L r d L+ r, g 3 =d L+ r u L r, and g =d L r u L+ r, we can construct four wave functions L+ = g + g + g 3 + g /, L+ = g g g 3 + g /, 3 L+ = g + g g 3 g /, L+ = g g + g 3 g /, of which corresponds to the odd-parity spin singlet and to the even-parity singlet, while 3 and are spin triplets of the even and odd parities, respectively. Energy levels of and 3 will be degenerate due to the electron separation Eq. 5. For the same reason, and will correspond to the same energy. The electrons described by wave functions and tend to avoid one another, i.e., to stay at opposite angles see below. On the other hand, in states and 3 the electrons tend to be localized one above the other at the same angle. States of these symmetries never correspond to the ground state and will not be discussed in the following. 3 The exact two-electron eigenfunctions for a given total angular momentum L are strictly independent of the flux. This is due to the following: no influence of the magnetic flux on the single-electron wave functions and hence on the Coulomb matrix elements, and the single-electron energies of all the configurations resulting in the same total angular momentum L are shifted by the magnetic flux by the same offset L / + / /mr. Moreover, the energy spectrum of the electron couple remains perfectly periodic with the flux-induced angular momentum transitions, since the Coulomb matrix elements depend only on the differences of the single-electron angular momenta and the increments resulting from the AB effect exactly cancel when the total L increases. Summarizing, due to the spin and parity degeneracies discussed above, for fixed ring parameters R,Z the interring electron-electron correlation will depend uniquely on the parity of the ground-state L quantum number

3 PHYSICAL REVIEW B 75, 3533 共7兲 ELECTRON CORRELATIONS IN CHARGE COUPLED E [mev]. R=3. nm Z=3R.5 which results from the symmetry and does not vanish even in the Z limit. The angular momentum eigenstates always have a rotationally invariant charge density. To inspect the electron-electron correlation, we use the pair-correlation function 共PCF兲 L= L=3 3.5 PCF共ra,rb兲 = L= L= 3. interaction Φ/Φ FIG.. 共Color online兲 Exact energy levels of given angular momentum L for R = 3. nm and Z = 3R are plotted in solid lines. The solid line at the bottom of the plot shows the expectation value of the electron-electron interaction energy calculated for the exact ground state. Dotted lines show the results of the Hartree approximation for the total ground-state energy 共the higher line, very close to the exact energy兲 and the interaction energy 共the lower line兲. The inset shows the zoom of the region of the near-threefold degeneracy at = /. III. RESULTS A. Exact solution Figure shows the energy spectrum for the smaller of the considered radii of R = 3. nm at a relatively large distance of Z = 3R 共the exact energy levels are plotted with solid lines, with total L indicated兲. At odd multiples of half of the flux quantum, the ground state is nearly threefold degenerate with respect to the angular momentum. The strict threefold degeneracy appears in the noninteracting case. However, the quantum-mechanical expectation value of the interaction energy 共plotted with the solid line at the bottom of the figure兲 is nonzero, and the observed near-three fold degeneracy at odd multiples of / results from an almost constant interaction for both even and odd L eigenstates. The interaction is slightly weaker at odd L, which lifts the threefold degeneracy and leads to a short appearance of odd L in the ground state 共see the dips of the interaction energy plot兲. The smaller value of the interaction energy in the states of odd L is due to the electron-electron interring correlation, drdr兩 共r,r兲兩关 3共ra r兲 3共rb r兲 + 3共ra r兲 3共rb r兲兴, L=. 冕 共兲 which gives the probability density to find one of the electrons at position ra and the other at rb. For illustrations, we fix ra in the lower ring 共za = Z / 兲 at an angle a = and observe the PCF as a function of the angle = b in the upper ring. In the weak-interaction 共large Z兲 limit, at even angular momenta the pair-correlation function for both + and 关Eq. 共兲兴 equals PCF共 a, b兲 = 兩dL共ra兲uL共rb兲兩 / = 共 兲, showing no signs of any dependence on a b, the electron positions being, therefore, perfectly uncorrelated. On the other hand, for states with odd angular momenta that are described by functions and, one obtains PCF共 a, b兲 = 共 兲关 cos共 a b兲兴; i.e., in these wave functions, the electrons are forbidden to stay at the same angle. The numerical results for the ground-state paircorrelation function are displayed in Fig. 共a兲 as functions of the external flux. For odd L 共near odd multiples of / 兲, the exact PCF is well approximated by the above analytical expression, and the calculated minimum value at = is exactly zero, even when more configurations contribute to the ground state. For even L, the pair-correlation function is nearly constant but a shallow minimum does appear at = due to nonzero contribution of configurations different than those used in Eq. 共兲 for construction of the approximate wave function. The total energy for smaller interring distance Z = R is plotted with the upper solid line in Fig. 3. Compared to Z = 3R, odd angular momenta appear in the ground state for larger flux intervals near odd multiples of /. The contour plot for the corresponding PCF is given in Fig. 共a兲. The correlation in the even L ground states is noticeably increased with respect to Z = 3R case. When the interring distance is decreased even further, the pair-correlation plots in odd and even L states become similar; see Fig. 共c兲 for Z =.5R and Fig. 共d兲 for Z =.35R. The PCF evolution with Z for even and odd angular momenta is plotted in Figs. 5共a兲 and 5共b兲 for R = 3. nm and in Figs..... R=3.nm Z=3R FIG.. 共Color online兲 共a兲 The exact pair-correlation function plotted for another electron fixed in the lower ring at the angle = as function of the magnetic flux for R = 3. nm and Z = 3R. 共b兲 Hartree charge density of the electron localized in the upper ring as obtained in the Hartree solution. 共c兲 Same as 共b兲 but for the electron localized in the lower ring

4 SZAFRAN, BEDNAREK, AND DUDZIAK E[meV] total L= L= R=3. nm Z=R total (Hartree)..5.. Φ/Φ 5 c and 5 d for R=3 nm. PCFs for odd and even L become identical in the Z= limit of a single ring, in which the one-dimensional Coulomb interaction prevents the electrons to appear at the same angle. For the small ring R =3. nm in odd L eigenstates, the Coulomb hole in the PCF plot formed at = due to the symmetry is only slightly increased when passing through the Z= limit see Fig. 5 b. For odd L states in the larger ring R=3 nm, the size of the Coulomb hole grows in a more pronounced manner in the Z= limit Fig. 5 d. The PCF for large Z is independent of R, but the dependence on R for Z= is evident compare Figs. 5 a and 5 b with Figs. 5 c and 5 d near Z=. Solid lines in Fig. show the PCF for both electrons localized at the same angle a = b in the even L ground states as function of the plots correspond to the cross section of the contour plots in Figs. 5 a and 5 c calculated for a = ; for odd L the PCF a = b is strictly. For R =3 nm, the probability to find one electron above the other at the same angle becomes negligible already around Z =R, but for the strong confinement R=3. nm the PCF vanishes only in the Z= limit. B. Mean-field picture In the Hartree approach, we look for a separable wave function H = R z Z/ R z + Z/ minimizing the expectation value of Hamiltonian, which reduces to the self-consistent solution of the system of eigenequations e h + d = e, r L= interaction L=3 e L= FIG. 3. Color online Same as Fig. but for Z=R. The dashed curve marked by e is the mean-field single-electron energy, the eigenvalue of Eqs. and 3. Z=R Φ/Φ Φ/Φ (d) Z=.5R.5.5 Φ/Φ Φ/Φ (e) (f) Z=.35R.5.5 Φ/Φ Φ/Φ. e h + d = e, 3 r where e and e are the single-electron energies always equal when the self-consistency is reached. We allow the functions and to deviate from angular momentum eigenstates and thus the corresponding charge densities from being perfectly circular. The angular wave functions are calculated with the imaginary time technique 5 and a random charge distribution is introduced at the start of the selfconsistent iteration. The circular symmetry of the charge distribution is restored during the iteration whenever it corresponds to the best mean-field approximation. The Hartree interaction energy calculated as E int =. PHYSICAL REVIEW B 75, FIG.. Color online a, c, and e Contour plots of the exact pair-correlation function for an electron position fixed in the lower ring at the angle =. The numbers inside the plots give the ground-state angular momentum. The scale for PCF multiplied by is given by the horizontal bar at the bottom of the plot. b, d, and f Hartree charge density of the electron localized in the upper ring. In all the plots R=3. nm, and the lightest shade corresponds to the minimal value of the presented functions. e d d r is used to evaluate the total energy of the system E tot = e + e E int

5 ELECTRON CORRELATIONS IN CHARGE COUPLED PHYSICAL REVIEW B 75, even L odd L.5 R=3. nm R=3 nm.5 even L odd L.5 (d).5.5 (e) Φ= Φ= (g) R=3. nm R=3 nm (f) (h) Φ=.5Φ Φ=.5Φ.5.5 FIG. 5. Color online a d Contour plots of the exact pair-correlation function for an electron position fixed in the lower ring at the angle = as function of the interdot distance for a and b R=3. nm and c and d R=3 nm. Plots a and c correspond to the even L states, and b and d to the odd L states. e g Contour plots of the ground-state Hartree charge density of the electron in the upper ring as function of the interdot distance for e and f R=3. nm and g and h R=3 nm. Plots e and g correspond to fluxes equal to even multiples of /, and f and h to odd multiples of /. Note the inversion of the vertical axis direction in upper and lower panels. The magnetic-field dependence of the mean-field energy on the magnetic field is plotted in Fig. by the dotted line for R=3. nm. The Hartree total and interaction energies are nearly equal to the exact energies in the ground state when it corresponds to even L, but near the odd multiples of half of the flux quantum, the Hartree total energy slightly overestimates the ground-state energy. At the angular momentum transitions, the exact interaction energy is a discontinuous function of only the exact total energy is continuous. On the other hand, the Hartree interaction energy is continuous as a function of the flux. The minima of the Hartree interaction energy at odd multiples of / are shallower than for the exact interaction energy. The pair-correlation function calculated for the mean-field wave function H Eq. is PCF, =. Therefore, all the correlation that Hartree approach takes into account is expressed by the charge densities of the two electrons. When the single-electron wave functions and are the eigenfunctions of the singleelectron angular momentum, the interelectron correlation is completely overlooked since the charge densities are rotationally invariant. The Hartree electron densities in the upper and lower rings for R=3. nm and Z=3R are plotted in Figs. b and c, respectively. We see that the mean-field charge densities are rotationally invariant, with the exception of the magnetic fluxes nearly equal to odd multiples of /. In that case, the rotational symmetry is broken and the obtained probability densities are shifted from one another by a angle = +. The plot for the upper ring is very similar to the exact PCF presented in Fig. a. The difference is that the exact PCF for even L is only approximately rotationally invariant, while the invariance of the corresponding Hartree result is strict. The comparison of the exact ground-state energy to the mean-field approximation for Z=R is presented in Fig. 3. We noticed that the ground-state energy is noticeably overestimated by the Hartree approach for all. The ground-state energy possesses cusps at the ground-state angular momentum transitions, but the Hartree energy is a smooth function of the flux. However, the cusps are observed in the singleelectron Hartree energy and appear when the circular symmetry of the Hartree charge density is broken. Note that the flux range corresponding to the broken-symmetry mean-field solution is wider than the one in which the odd L ground state is realized in the exact solution. The contour plot of the charge density in the upper ring for Z=R is plotted in Fig. b. Both the mean field and exact Fig. a solutions indicate an increase of the angular electron-electron correlation near odd multiples of / a larger Coulomb hole and a π PCF (φ =φ )..5 R=3.nm R=3nm Φ=nΦ..5.. min (πρ) FIG.. Color online Solid lines The pair-correlation function for equal localization angles of both electrons as function of the interdot distance for the exact ground state of even L cross section of Figs. 5 a and 5 c at =. Dotted lines The minimal value of the Hartree charge density calculated for integer fluxes cross sections of Figs. 5 e and 5 g

6 PHYSICAL REVIEW B 75, 3533 共7兲 SZAFRAN, BEDNAREK, AND DUDZIAK total single electron interaction..3.. FIG. 7. 共Color online兲 Results of the mean-field approach for R = 3. nm as function of the external flux and the interring separation: 共a兲 Deviation of the single-electron charge density from the uniform value Eq. 共兲, 共b兲 interaction energy, 共c兲 single-electron energy, and 共d兲 the total energy. more probable localization of the other electron near a = b + 兲. In contrast to the exact solution, the mean-field approximation yields a continuous dependence of the interaction energy on the flux. For smaller value of Z, the flux ranges in which the Hartree charge density is rotationally invariant disappear. The flux dependence of the charge density is plotted in Figs. 共d兲 and 共f兲 for Z =.5R and Z =.35R, respectively. The Coulomb hole in the Hartree charge density distribution continuously pulsates with the external flux and acquires maximal angular width near odd multiples of /. The Hartree charge density corresponding to the maximal size of the Coulomb hole 共odd multiples of / 兲 is plotted in Figs. 5共f兲 and 5共h兲 and that compared to the charge density for integer multiples of 共for which the Coulomb hole, if any, has a minimal angular width兲 is presented in Figs. 5共e兲 and 5共g兲. Supplementary illustrations of the minimal charge density for integer fluxes are given by the dotted lines in Fig.. The Coulomb hole at integer fluxes appears abruptly at Z.7R for R = 3. nm and at Z.R for R = 3 nm. At the critical Z values, the symmetry breaking changes its character from reentrant, i.e., appearing only near odd multiples of /, to pulsating stronger or weaker, but appearing at an arbitrary flux. The pulsations of the charge density acquire smaller amplitude in the Z = limit 关see also Fig. 共f兲兴. The deviation of the single-electron charge density from rotational symmetry 共for which the 兩 共 兲兩 = 兲 can be conveniently estimated by the mean value 冕冉 兩 共 兲兩 冊 d (d) = R=3 nm Hartree - exact [mev] energy [mev].5 共兲 The dependence on the external flux and the interring distance is presented by the surface plot in Fig. 7共a兲 for R = 3. nm. At large Z, falls to zero off the proximity of fractional fluxes. Passage from the reentrant to persistent symmetry breaking as function of the flux appears when the valleys of = end: near Z / R =.7 and integer fluxes. The FIG.. The mean-field energy contributions for R = 3 nm at integer flux quanta. The inset shows the total energy overestimated by the Hartree approach. deviations off the rotational symmetry are translated into lowered interaction energy given in Fig. 7共b兲. Note that the interaction energy is maximal at integer fluxes when the flat = valleys end. The discontinuities in the derivative of the interaction energy are translated into cusps of the singleelectron interaction energy 关Fig. 7共c兲兴, but the derivative discontinuities cancel in Eq. 共5兲 and, in consequence, the total Hartree energy 关Fig. 7共d兲兴 has a smooth surface. Note that in the Z = limit the dependence of all the quantities on disappears. Figure demonstrates that for R = 3 nm, the Hartree interaction energy approach the Z = limit monotonically. No local maximum as the one observed for integer fluxes for stronger confinement 关see Fig. 7共b兲兴 is observed. At integer fluxes and Z.R, when the mean-field solution has rotational symmetry 关see Figs. 5共g兲 and 兴, the single-electron and total energies become equal to the interaction energy 共then the single-electron states are the angular momentum L eigenfunctions 关cf. Eq. 共兲兴, with zero kinetic energy at = L, hence e = e = Eint兲. Below Z.R, when the broken-symmetry solution appears, the three energies split. For fractional fluxes, the rotational symmetry is always broken and the energies discussed are split for arbitrary Z. The variational overestimate of the exact energy by the Hartree solution for R = 3 nm is displayed in the inset of Fig.. We see that the overestimate is largest in the singlering limit and tends to zero at large interring distance. Finally, in Fig. 9, we present the mean-field expectation value of the angular momentum as compared to the exact ground-state quantum number for R = 3. nm and various Z. Figure 9共a兲 corresponds to the reentrant symmetry breaking. For the conserved rotational symmetry, the Hartee angular momentum is equal to the exact even-parity values of L. In the intervals of the broken-symmetry Hartree solution, which for that case corresponds well to the odd L exact ground states, the Hartree angular momentum changes linearly. Figure 9共b兲 shows the critical case of Z =.7R when the symmetry breaking becomes persistent. The Hartree angular momentum has zero derivatives at the integer fluxes for which the exact L is even. For the small interring distance of Z =.R, the Hartree angular momentum becomes linear with 3533-

7 ELECTRON CORRELATIONS IN CHARGE COUPLED L L L..5.. Z= R Φ/Φ..5.. Z=.7R Z=.R Φ/Φ..5.. Φ/Φ FIG. 9. Color online The exact ground-state angular momentum solid lines and the mean-field expectation value dotted line for R=3. nm and a Z=R, b Z=.7R, and c Z=.R.. Exact values of the angular momentum are reproduced in the Hartree approach at integer multiples of /. The last property is obtained independent of Z. Let us briefly discuss the symmetry breaking obtained in the mean-field method. The wave function in the Hartree approach can only reproduce one of the features of the exact solution: either the quantized angular momentum eigenvalue implying circular symmetry of the charge density or the correlation between angular positions of the electrons which is accompanied by the broken rotational symmetry of the charge density in both rings. The symmetry breaking at large Z has a reentrant character; it reappears in the intervals near odd multiples of /, outside of which the Hartree density possesses perfectly circular symmetry. Reentrant pinning of Wigner molecules induced by an external charge defect in otherwise circular quantum dots was reported previously in the exact calculations. The pinning is due to mixing of angular momentum states and results in the extraction of the molecular electron distribution from the internal coordinates to the laboratory frame. The reentrant symmetry breaking presented in this work appears in the approximate solution and is self-induced spontaneous ; i.e., the role of the external defect breaking the symmetry of the potential is played by the charge density of the other electron. As the interring distance decreases, the flux intervals of perfect circular symmetry of the Hartree charge density become shorter, and eventually disappear. Wigner molecules formed in the exact solution in the internal coordinates 3 with conserved rotational symmetry of the charge density, in the PHYSICAL REVIEW B 75, Hartree approach, appear in the laboratory frame. The energy overestimate see the inset of Fig. obtained in the Hartree method in the Z= limit is due to an additional kinetic energy that appears due to the localization of the singleelectron charge islands that are present in the brokensymmetry solution and to only partial accounting of the correlation taken by the Hartree method. In the Hartree method, the electrons do not react on their actual relative positions; only the entire charged islands do. In circular dots, the part of correlation neglected by the mean-field approach is compensated by an exaggerated localization 7 of the electron islands with respect to the exact broken-symmetry states constructed at the exact ground-state degeneracy 7. One also notices this feature in vertically stacked quantum rings: the angular size of the Coulomb hole obtained in the broken symmetry states is larger than in the exact solution see Figs. and 5. In one-dimensional quantum dots, the Wigner crystallization appears for long structures in the laboratory frame of reference.,9 The Hartree-Fock approach does not have to break the symmetry to reproduce the laboratory-frame Wigner crystallization, and the overlooked correlation becomes negligible in long dot limit. 9 In consequence, the Hartree-Fock energy becomes exact 9 in the Wigner crystallization limit of one-dimensional dots in contrast to the system discussed in this paper. IV. SUMMARY AND CONCLUSIONS We have studied an electron pair in a couple of stacked quantum rings assuming a purely electrostatic coupling by means of exact and mean-field methods. The spatial correlation between the electrons, i.e., the extent in which they avoid staying one above the other, undergoes oscillations as function of the magnetic flux. For the ground states of odd angular momentum, the electrons are, by symmetry constraints, strictly forbidden to stay at the same angle. This is not the case for even angular momentum ground states, for which the electron tendency to stay at the opposite angles is not required by the symmetry and vanishes in the large interring distance. The oscillations of the pair-correlation function with the external flux decrease in amplitude when the rings get closer, i.e., when Wigner molecules are formed in both even and odd angular momentum states. The mean-field solution at large interring distance breaks the rotational symmetry in a reentrant fashion near odd multiples of the flux quantum to mimic the electron distribution in the exact odd angular momentum states. Although the range of the odd angular momentum ground-state stability shrinks with the interring distance, the ground state conserves an odd angular momentum at odd multiples of half of the flux quantum for arbitrarily far situated rings. ACKNOWLEDGMENT This paper was supported by the Polish State Committee for Scientific Research KBN under Grant No. P3B

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